27edo
| ← 26edo | 27edo | 28edo → |
27 equal divisions of the octave (27edo), or 27(-tone) equal temperament (27tet, 27et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 27 equally large steps. Each step represents a frequency ratio of the 27th root of 2, or 44.4 cents.
Theory
If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444… cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics 3, 5, and 7 sharply. The optimal step size for octave-shrinking in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614-edo, which corresponds to a step size of 44.3023 cents.
Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13+2/3 cents. The result is that 6/5, 7/5 and especially 7/6 are all tuned more accurately than this. It can be considered the superpythagorean counterpart of 19edo, as its 5th is audibly indistinguishable from 1/3 septimal comma superpyth in the same way that 19edo is audibly indistinguishable from 1/3 syntonic comma meantone, resulting in three of them reaching a near perfect minor third/major sixth in both.
27edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also). These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with 22edo tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support the superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.
Though the 7-limit tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both consistently and distinctly – that is, everything in the 7-odd-limit diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament. It also approximates 19/10, 19/12, and 19/14, so 0-7-13-25 does quite well as a 10:12:14:19; the major seventh 25\27 is less than a cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making it the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest harmonic entropy possible and thus is, in theory, most dissonant, assuming the relatively common values of a = 2 and s = 1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
The 27-note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 edosteps, just one edostep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.2 | +13.7 | +9.0 | +18.3 | -18.0 | +3.9 | -21.6 | -16.1 | +13.6 | +18.1 | -6.1 |
| Relative (%) | +20.6 | +30.8 | +20.1 | +41.2 | -40.5 | +8.8 | -48.6 | -36.1 | +30.6 | +40.7 | -13.6 | |
| Steps (reduced) |
43 (16) |
63 (9) |
76 (22) |
86 (5) |
93 (12) |
100 (19) |
105 (24) |
110 (2) |
115 (7) |
119 (11) |
122 (14) | |
Intervals
| # | Cents | Approximate Ratios* | Ups and Downs Notation | 6L 1s notation | Solfege | ||||
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.00 | 1/1 | P1 | perfect unison | D | perfect unison | C | do | |
| 1 | 44.44 | 28/27, 36/35, 39/38, 49/48, 50/49, 81/80 | ^1, m2 | up unison, minor 2nd | ^D, Eb | aug 1sn, double-dim 2nd | C#, Dbbb | di | |
| 2 | 88.89 | 16/15, 21/20, 25/24, 19/18, 20/19 | ^^1, ^m2 | dup unison, upminor 2nd | ^^D, ^Eb | double-aug 1sn, dim 2nd | Cx, Dbb | ra | |
| 3 | 133.33 | 15/14, 14/13, 13/12 | vA1, ~2 | downaug 1sn, mid 2nd | vD#, vvE | triple-aug 1sn, minor 2nd | Cx#, Db | ru | |
| 4 | 177.78 | 10/9 | A1, vM2 | aug 1sn, downmajor 2nd | D#, vE | major 2nd | D | reh | |
| 5 | 222.22 | 8/7, 9/8 | M2 | major 2nd | E | aug 2nd, double-dim 3rd | D#, Ebbb | re | |
| 6 | 266.67 | 7/6 | m3 | minor 3rd | F | double-aug 2nd, dim 3rd | Dx, Ebb | ma | |
| 7 | 311.11 | 6/5, 19/16 | ^m3 | upminor 3rd | Gb | triple-aug 2nd, minor 3rd | Dx#, Eb | me | |
| 8 | 355.56 | 16/13 | ~3 | mid 3rd | ^Gb | E | major 3rd | E | mu |
| 9 | 400.00 | 5/4, 24/19 | vM3 | downmajor 3rd | vF# | aug 3rd, double-dim 4th | E#, Fbbb | mi | |
| 10 | 444.44 | 9/7, 13/10 | M3 | major 3rd | F# | double-aug 3rd, dim 4th | Ex, Fbb | mo | |
| 11 | 488.89 | 4/3 | P4 | perfect 4th | G | triple-aug 3rd, minor 4th | Ex#, Fb | fa | |
| 12 | 533.33 | 27/20, 48/35, 19/14, 26/19 | ^4 | up 4th | Ab | major 4th | F | fih | |
| 13 | 577.78 | 7/5, 18/13 | ~4, ^d5 | mid 4th, updim 5th | ^^G, ^Ab | aug 4th, double-dim 5th | F#, Gbbb | fi | |
| 14 | 622.22 | 10/7, 13/9 | vA4, ~5 | downaug 4th, mid 5th | vG#, vvA | double-aug 4th, dim 5th | Fx, Gbb | se | |
| 15 | 666.67 | 40/27, 35/24, 19/13, 28/19 | v5 | down fifth | G# | triple-aug 4th, minor 5th | Fx#, Gb | sih | |
| 16 | 711.11 | 3/2 | P5 | perfect 5th | A | major 5th | G | so/sol | |
| 17 | 755.56 | 14/9, 20/13 | m6 | minor 6th | Bb | aug 5th, double-dim 6th | G#, Abbb | lo | |
| 18 | 800.00 | 8/5, 19/12 | ^m6 | upminor 6th | ^Bb | double-aug 5th, dim 6th | Gx, Abb | le | |
| 19 | 844.44 | 13/8 | ~6 | mid 6th | vA# | triple-aug 5th, minor 6th | Gx#, Ab | lu | |
| 20 | 888.89 | 5/3, 32/19 | vM6 | downmajor 6th | A# | major 6th | A | la | |
| 21 | 933.33 | 12/7 | M6 | major 6th | B | aug 6th, double-dim 7th | A#, Bbbb | li | |
| 22 | 977.78 | 7/4, 16/9 | m7 | minor 7th | C | double-aug 6th, dim 7th | Ax, Bbb | ta | |
| 23 | 1022.22 | 9/5 | ^m7 | upminor 7th | Db | triple-aug 6th, minor 7th | Ax#, Bb | te | |
| 24 | 1066.67 | 28/15, 13/7, 24/13 | ~7 | mid 7th | ^Db | major 7th | B | tu | |
| 25 | 1111.11 | 15/8, 40/21, 48/25, 19/10, 36/19 | vM7 | downmajor 7th | vC# | aug 7th, double-dim 8ve | B#, Cbb | ti | |
| 26 | 1155.56 | 27/14, 35/18, 96/49, 49/25, 160/81 | M7 | major 7th | C# | double-aug 7th, dim 8ve | Bx, Cb | da | |
| 27 | 1200.00 | 2/1 | P8 | 8ve | D | 8ve | C | do | |
* based on treating 27edo as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color name | monzo format | examples |
|---|---|---|---|
| minor | zo | {a, b, 0, 1} | 7/6, 7/4 |
| fourthward wa | {a, b}, b < -1 | 32/27, 16/9 | |
| upminor | gu | {a, b, -1} | 6/5, 9/5 |
| mid | tho | {a, b, 0, 0, 0, 1} | 13/12, 13/8 |
| thu | {a, b, 0, 0, 0, -1} | 16/13, 24/13 | |
| downmajor | yo | {a, b, 1} | 5/4, 5/3 |
| major | fifthward wa | {a, b}, b > 1 | 9/8, 27/16 |
| ru | {a, b, 0, -1} | 9/7, 12/7 |
All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:
| color of the 3rd | JI chord | notes as edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| zo | 6:7:9 | 0-6-16 | C Eb G | Cm | C minor |
| gu | 10:12:15 | 0-7-16 | C ^Eb G | C^m | C upminor |
| ilo | 18:22:27 | 0-8-16 | C vvE G | C~ | C mid |
| yo | 4:5:6 | 0-9-16 | C vE G | Cv | C downmajor or C down |
| ru | 14:18:21 | 0-10-16 | C E G | C | C major or C |
For a more complete list, see Ups and Downs Notation #Chords and Chord Progressions. See also the 22edo page.
JI approximation
15-odd-limit interval mappings
The following table shows how 15-odd-limit intervals are represented in 27edo. Prime harmonics are in bold; inconsistent intervals are in italic.
| Interval, complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 7/6, 12/7 | 0.204 | 0.5 |
| 15/11, 22/15 | 3.617 | 8.1 |
| 13/8, 16/13 | 3.917 | 8.8 |
| 5/3, 6/5 | 4.530 | 10.2 |
| 9/5, 10/9 | 4.626 | 10.4 |
| 7/5, 10/7 | 4.734 | 10.7 |
| 13/7, 14/13 | 5.035 | 11.3 |
| 13/12, 24/13 | 5.239 | 11.8 |
| 11/9, 18/11 | 8.148 | 18.3 |
| 7/4, 8/7 | 8.952 | 20.1 |
| 3/2, 4/3 | 9.156 | 20.6 |
| 9/7, 14/9 | 9.360 | 21.1 |
| 13/10, 20/13 | 9.770 | 22.0 |
| 11/10, 20/11 | 12.774 | 28.7 |
| 5/4, 8/5 | 13.686 | 30.8 |
| 15/14, 28/15 | 13.891 | 31.3 |
| 13/9, 18/13 | 14.395 | 32.4 |
| 11/6, 12/11 | 17.304 | 38.9 |
| 11/7, 14/11 | 17.508 | 39.4 |
| 11/8, 16/11 | 17.985 | 40.5 |
| 9/8, 16/9 | 18.312 | 41.2 |
| 15/13, 26/15 | 18.926 | 42.6 |
| 15/8, 16/15 | 21.602 | 48.6 |
| 13/11, 22/13 | 21.901 | 49.3 |
The following tables show how 15-odd-limit intervals are represented in 27edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 7/6, 12/7 | 0.204 | 0.5 |
| 15/11, 22/15 | 3.617 | 8.1 |
| 13/8, 16/13 | 3.917 | 8.8 |
| 5/3, 6/5 | 4.530 | 10.2 |
| 9/5, 10/9 | 4.626 | 10.4 |
| 7/5, 10/7 | 4.734 | 10.7 |
| 13/7, 14/13 | 5.035 | 11.3 |
| 13/12, 24/13 | 5.239 | 11.8 |
| 11/9, 18/11 | 8.148 | 18.3 |
| 7/4, 8/7 | 8.952 | 20.1 |
| 3/2, 4/3 | 9.156 | 20.6 |
| 9/7, 14/9 | 9.360 | 21.1 |
| 13/10, 20/13 | 9.770 | 22.0 |
| 11/10, 20/11 | 12.774 | 28.7 |
| 5/4, 8/5 | 13.686 | 30.8 |
| 15/14, 28/15 | 13.891 | 31.3 |
| 13/9, 18/13 | 14.395 | 32.4 |
| 11/6, 12/11 | 17.304 | 38.9 |
| 11/7, 14/11 | 17.508 | 39.4 |
| 11/8, 16/11 | 17.985 | 40.5 |
| 9/8, 16/9 | 18.312 | 41.2 |
| 15/13, 26/15 | 18.926 | 42.6 |
| 15/8, 16/15 | 21.602 | 48.6 |
| 13/11, 22/13 | 21.901 | 49.3 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 7/6, 12/7 | 0.204 | 0.5 |
| 13/8, 16/13 | 3.917 | 8.8 |
| 5/3, 6/5 | 4.530 | 10.2 |
| 9/5, 10/9 | 4.626 | 10.4 |
| 7/5, 10/7 | 4.734 | 10.7 |
| 13/7, 14/13 | 5.035 | 11.3 |
| 13/12, 24/13 | 5.239 | 11.8 |
| 7/4, 8/7 | 8.952 | 20.1 |
| 3/2, 4/3 | 9.156 | 20.6 |
| 9/7, 14/9 | 9.360 | 21.1 |
| 13/10, 20/13 | 9.770 | 22.0 |
| 5/4, 8/5 | 13.686 | 30.8 |
| 15/14, 28/15 | 13.891 | 31.3 |
| 13/9, 18/13 | 14.395 | 32.4 |
| 11/8, 16/11 | 17.985 | 40.5 |
| 9/8, 16/9 | 18.312 | 41.2 |
| 15/13, 26/15 | 18.926 | 42.6 |
| 13/11, 22/13 | 21.901 | 49.3 |
| 15/8, 16/15 | 22.842 | 51.4 |
| 11/7, 14/11 | 26.936 | 60.6 |
| 11/6, 12/11 | 27.141 | 61.1 |
| 11/10, 20/11 | 31.671 | 71.3 |
| 11/9, 18/11 | 36.297 | 81.7 |
| 15/11, 22/15 | 40.827 | 91.9 |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [43 -27⟩ | [⟨27 43]] | -2.89 | 2.88 | 6.50 |
| 2.3.5 | 128/125, 20000/19683 | [⟨27 43 63]] | -3.88 | 2.74 | 6.19 |
| 2.3.5.7 | 64/63, 126/125, 245/243 | [⟨27 43 63 76]] | -3.70 | 2.39 | 5.40 |
| 2.3.5.7.13 | 64/63, 91/90, 126/125, 169/168 | [⟨27 43 63 76 100]] | -3.18 | 2.39 | 5.39 |
| 2.3.5.7.13.19 | 64/63, 76/75, 91/90, 126/125, 169/168 | [⟨27 43 63 76 100 115]] | -3.18 | 2.18 | 4.92 |
27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next equal temperaments doing better in those subgroups are 31, 31, and 46, respectively.
27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next equal temperament that does better in this subgroup is 53.
Rank-2 temperaments
| Periods per 8ve |
Generator | Temperaments | MOS Scales |
|---|---|---|---|
| 1 | 1\27 | Quartonic/quarto | |
| 1 | 2\27 | Octacot/octocat | 1L_12s, 13L_1s |
| 1 | 4\27 | Tetracot/modus/wollemia | 1L_5s, 6L_1s, 7L_6s, 7L_13s |
| 1 | 5\27 | Machine/kumonga | 1L_4s, 5L_1s, 5L_6s, 11L_5s |
| 1 | 7\27 | Myna/coleto/minah/oolong | 4L_3s, 4L_7s, 4L_11s, 4L_15s, 4L_19s |
| 1 | 8\27 | Beatles/ringo | 3L_4s, 7L_3s, 10L_7s |
| 1 | 10\27 | Sensi/sensis | 3L_2s, 3L_5s, 8L_3s, 8L_11s |
| 1 | 11\27 | Superpyth | 5L_2s, 5L_7s, 5L_12s, 5L_17s |
| 1 | 13\27 | Fervor | 2L_3s, 2L_5s, 2L_7s, 2L_9s, 2L_11s, etc ... 2L_23s |
| 3 | 1\27 | Semiaug/hemiaug | |
| 3 | 2\27 | Augmented/Augene/ogene | 3L_3s, 3L_6s, 3L_9s, 12L_3s |
| 3 | 4\27 | Oodako/terrain | 3L_3s, 6L_3s, 6L_9s, 6L_15s |
| 9 | 1\27 | Terrible version of Ennealimmal /niner |
9L_9s |
Commas
27edo tempers out the following commas. (Note: This assumes the val ⟨27 43 63 76 93 100].)
| Prime Limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 5 | 128/125 | [7 0 -3⟩ | 41.06 | Trigu | Diesis, augmented comma |
| 5 | 20000/19683 | [5 -9 4⟩ | 27.66 | Saquadyo | Minimal diesis, Tetracot comma |
| 5 | 78732/78125 | [2 9 -7⟩ | 13.40 | Sepgu | Medium semicomma, Sensipent comma |
| 5 | (26 digits) | [1 -27 18⟩ | 0.86 | Satritribiyo | Ennealimma |
| 7 | 686/675 | [1 -3 -2 3⟩ | 27.99 | Trizo-agugu | Senga |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal comma, Archytas' comma, Leipziger Komma |
| 7 | 50421/50000 | [-4 1 -5 5⟩ | 14.52 | Quinzogu | Trimyna |
| 7 | 245/243 | [0 -5 1 2⟩ | 14.19 | Zozoyo | Sensamagic |
| 7 | 126/125 | [1 2 -3 1⟩ | 13.79 | Zotrigu | Septimal semicomma, Starling comma |
| 7 | 4000/3969 | [5 -4 3 -2⟩ | 13.47 | Rurutriyo | Octagar |
| 7 | 1728/1715 | [6 3 -1 -3⟩ | 13.07 | Triru-agu | Orwellisma, Orwell comma |
| 7 | (12 digits) | [-6 -8 2 5⟩ | 1.12 | Quinzo-ayoyo | Wizma |
| 7 | 2401/2400 | [-5 -1 -2 4⟩ | 0.72 | Bizozogu | Breedsma |
| 7 | 4375/4374 | [-1 -7 4 1⟩ | 0.40 | Zoquadyo | Ragisma |
| 7 | (12 digits) | [-4 6 -6 3⟩ | 0.33 | Trizogugu | Landscape comma |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.58 | Loruru | Mothwellsma |
| 11 | 896/891 | [7 -4 0 1 -1⟩ | 9.69 | Saluzo | Pentacircle |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Lozoyo | Keenanisma |
| 13 | 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozogu | Superleap |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Music
- Music For Your Ears play – the central portion is in 27edo, the rest in 46edo.